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Showing 25 of 280 problems in Dynamical systems

NT-DYN-014

Collatz Conjecture

IconicConjecturec. 1937

Define

T : Z_{> 0} \to Z_{> 0}

T (n) = {n /2, 3 n + 1, n even, n odd .

For every

n \in Z_{> 0}

, there exists

k \geq 0

such that

T^{k} (n) = 1

NT-TRANS-018

Lehmer’s Mahler-Measure Conjecture

MajorExact constant problem1933

There exists a real constant

c > 1

such that every algebraic integer

α \neq = 0

that is not a root of unity satisfies

M (α) \geq c,

where, if

a_{0} \prod_{j = 1}^{d} (x - α_{j}) \in Z [x]

is the minimal polynomial of

α

, then

M (α) = ∣ a_{0} ∣ \prod_{j = 1}^{d} max (1, ∣ α_{j} ∣)

NT-DIOAPP-020

Littlewood Conjecture

MajorConjecturec. 1930

For all real numbers

α, β

n \to \infty lim inf n ∥ n α ∥ ∥ n β ∥ = 0,

where

∥ x ∥ = min_{m \in Z} ∣ x - m ∣

ALG-GROUP-021

Soficity Conjecture

LandmarkConjecture2000

Every countable group is sofic. Explicitly, for every countable group

G

, finite

F \subseteq G

, and

ε > 0

, there are an integer

n

and a map

σ : G \to Sym (n)

such that

σ (1) = 1

d_{H} (σ (g h), σ (g) σ (h)) < ε (g, h \in F),

and

d_{H} (σ (g), 1) > 1 - ε

for every

g \in F ∖ {1}

, where

d_{H}

is normalized Hamming distance.

GEO-TOP-015

Weinstein conjecture

LandmarkConjecture1978

Let

M^{2 n - 1}

be a closed smooth manifold and let

α

be a contact form, so

α \land (d α)^{n - 1}

is nowhere zero. The Reeb vector field

R_{α}

, uniquely defined by

α (R_{α}) = 1

and

d α (R_{α}, \cdot) = 0

, has a nonconstant periodic orbit.

GEO-TOP-014

Nearby Lagrangian conjecture

MajorConjecturec. 1980

Let

Q

be a closed connected smooth manifold and

π : T^{*} Q \to Q

its cotangent projection. Define the canonical Liouville

1

-form by

λ_{(q, p)} (ξ) = p (d π_{(q, p)} ξ)

. If

L \subset T^{*} Q

is a closed connected Lagrangian submanifold and

λ ∣_{L} = df

for some smooth

f : L \to R

, then a compactly supported Hamiltonian isotopy of

T^{*} Q

carries

L

to the zero section.

PDE-002

Global regularity for three-dimensional incompressible Euler

LandmarkOpen problemc. 1930

For every divergence-free Schwartz vector field

u_{0} \in S (R^{3}; R^{3})

, the maximal classical solution of

\partial_{t} u + (u \cdot \nabla) u = - \nabla p, \nabla \cdot u = 0, u (\cdot, 0) = u_{0}

exists for all

t \geq 0

and belongs to

C^{\infty} (R^{3} \times [0, \infty))

PDE-005

Global regularity for the two-dimensional inviscid Boussinesq system

MajorOpen problemc. 1980

For every

u_{0} \in S (R^{2}; R^{2})

and

θ_{0} \in S (R^{2})

with

\nabla \cdot u_{0} = 0

, the maximal classical solution of

\partial_{t} u + (u \cdot \nabla) u + \nabla p = θ e_{2}, \partial_{t} θ + u \cdot \nabla θ = 0, \nabla \cdot u = 0, (u, θ) ∣_{t = 0} = (u_{0}, θ_{0})

exists for all

t \geq 0

and is smooth.

DYN-001

Uniform finiteness in Hilbert's sixteenth problem

IconicOpen problem1900

For every integer

d \geq 2

, there is an integer

H (d) < \infty

such that every planar polynomial vector field

\overset{x}{˙} = P (x, y), \overset{y}{˙} = Q (x, y), max {de g P, de g Q} \leq d,

having only finitely many limit cycles has at most

H (d)

limit cycles. A limit cycle is an isolated periodic orbit.

DYN-002

$C^r$ closing lemma

LandmarkConjecturec. 1960

Let

M

be a closed smooth manifold,

r \geq 2

f \in Diff^{r} (M)

, and

x \in M

a nonwandering point of

f

: every neighborhood

U ∋ x

has

f^{n} (U) \cap U \neq = \emptyset

for some

n > 0

. For every

C^{r}

neighborhood

U

f

, there is

g \in U

for which

g^{m} (x) = x

for some

m > 0

DYN-003

Density of hyperbolicity for rational maps

LandmarkConjecturec. 1920

For every

d \geq 2

, hyperbolic maps are dense in the space

Rat_{d}

of degree-

d

rational maps

f : C \to C

, with its coefficient topology. Here

f

is hyperbolic when the closure of the forward orbits of all its critical points is disjoint from its Julia set

J (f)

DYN-004

Mandelbrot local-connectivity conjecture

LandmarkConjecturec. 1982

The Mandelbrot set

M = {c \in C : {f_{c}^{n} (0) : n \geq 0} is bounded}, f_{c} (z) = z^{2} + c,

is locally connected: for every

c \in M

and every neighborhood

U

c

, there is a connected neighborhood

V

c

in the subspace

M

with

V \subset U

DYN-005

Birkhoff billiard conjecture

LandmarkConjecture1927

Let

Ω \subset R^{2}

be bounded with

C^{\infty}

, strictly convex boundary. Its billiard phase space is the open annulus

A = {(x, v) : x \in \partial Ω, ∣ v ∣ = 1, v points inward and is not tangent to \partial Ω}

, with the billiard map sending one reflected state to the next. If

A

is foliated by invariant circles homotopic to its boundary, then

\partial Ω

is an ellipse.

DYN-008

Positive metric entropy for the standard map

LandmarkOpen problemc. 1980

There exists

K \in R ∖ {0}

such that the area-preserving standard map

S_{K} (x, y) = (x + y + K sin (2 π x), y + K sin (2 π x)) (mod Z^{2})

T^{2}

has positive Kolmogorov--Sinai entropy with respect to normalized Lebesgue measure.

DYN-009

Furstenberg $\times2,\times3$ measure conjecture

LandmarkConjecture1967

Let

μ

be a Borel probability measure on

T = R / Z

that is invariant under

T_{2} (x) = 2 x (mod 1)

and

T_{3} (x) = 3 x (mod 1)

, and ergodic for their joint action: every Borel set invariant modulo

μ

under both maps has measure

0

1

. Then either

μ

is Lebesgue measure or there is a finite set

F \subset T

, invariant under both

T_{2}

and

T_{3}

, such that

μ (F) = 1

DYN-014

Sarnak Möbius disjointness conjecture

LandmarkConjecture2010

Define the Möbius function by

μ (1) = 1

μ (n) = 0

if a prime square divides

n

, and

μ (n) = (- 1)^{k}

n

is a product of

k

distinct primes. For every compact metric space

X

, every continuous map

T : X \to X

with zero topological entropy, every

f \in C (X; C)

, and every

x \in X

\frac{1}{N} n = 1 \sum N μ (n) f (T^{n} x) ⟶ 0 as N \to \infty.

DYN-006

Ivrii periodic-billiard conjecture

MajorConjecture1980

Let

Ω \subset R^{n}

be a bounded domain with

C^{\infty}

boundary, and let

B^{*} \partial Ω = {(x, ξ) \in T^{*} \partial Ω : ∣ ξ ∣ < 1}

, equipped with its canonical Liouville measure. The set of points of

B^{*} \partial Ω

that are periodic under the billiard map has measure zero.

DYN-007

Anosov-manifold conjecture

MajorConjecturec. 1970

Every closed connected smooth manifold that admits an Anosov diffeomorphism is homeomorphic to an infranilmanifold, namely a quotient

Γ\ N

, where

N

is a simply connected nilpotent Lie group and

Γ

is a torsion-free discrete subgroup of

N ⋊ C

for some compact subgroup

C \leq Aut (N)

, acting freely and cocompactly on

N

DYN-010

Rokhlin multiple-mixing problem

MajorOpen problem1949

Let

T

be an invertible measure-preserving transformation of a standard probability space

(X, B, μ)

. If

μ (A \cap T^{- n} B) \to μ (A) μ (B)

n \to \infty

for all

A, B \in B

, then for every

k \geq 2

, all

A_{0}, \dots, A_{k} \in B

, and all integer sequences

n_{1}^{(j)}, \dots, n_{k}^{(j)}

satisfying, as

j \to \infty

i min ∣ n_{i}^{(j)} ∣ \to \infty, i \neq = l min ∣ n_{i}^{(j)} - n_{l}^{(j)} ∣ \to \infty,

one has

μ (A_{0} \cap i = 1 ⋂ k T^{- n_{i}^{(j)}} A_{i}) ⟶ i = 0 \prod k μ (A_{i}) .

DYN-011

Banach simple-Lebesgue-spectrum problem

MajorOpen problem1932

There exist a standard nonatomic probability space

(X, B, μ)

and an invertible ergodic measure-preserving transformation

T : X \to X

such that the Koopman operator

U_{T} f = f \circ T

, restricted to

L_{0}^{2} (X, μ) = {f \in L^{2} (X, μ) : \int_{X} f d μ = 0},

is unitarily equivalent to the bilateral shift

S

ℓ^{2} (Z)

, defined by

S (e_{n}) = e_{n + 1}

DYN-012

Pugh–Shub stable-ergodicity conjecture

MajorConjecture1997

Let

M

be a closed connected manifold with a smooth probability volume

μ

, and fix a Riemannian norm on

TM

. For a linear map

A

, write

m (A) = in f_{∥ v ∥ = 1} ∥ A v ∥

. A

C^{2}

μ

-preserving diffeomorphism

f

is partially hyperbolic here if it has a continuous

D f

-invariant splitting into nonzero bundles

TM = E^{s} \oplus E^{c} \oplus E^{u}

and an integer

N \geq 1

such that, for every

x \in M

∥ D f_{x}^{N} ∣_{E_{x}^{s}} ∥ < m (D f_{x}^{N} ∣_{E_{x}^{c}}) \leq ∥ D f_{x}^{N} ∣_{E_{x}^{c}} ∥ < m (D f_{x}^{N} ∣_{E_{x}^{u}}), ∥ D f_{x}^{N} ∣_{E_{x}^{s}} ∥ < 1 < m (D f_{x}^{N} ∣_{E_{x}^{u}}) .

Among these diffeomorphisms, the stably ergodic ones are

C^{2}

-dense; stable ergodicity means that every sufficiently

C^{2}

-near

C^{2}

μ

-preserving diffeomorphism is ergodic.

DYN-013

Improbability of noncollision singularities

MajorConjecturec. 1984

Fix

N \geq 5

and masses

m_{1}, \dots, m_{N} > 0

. For

q_{i} (t) \in R^{3}

, consider

\overset{q}{¨}_{i} = j \neq = i \sum m_{j} \frac{q _{j} - q _{i}}{∣ q _{j} - q _{i} ∣ ^{3}}, i = 1, \dots, N,

and the collision-free phase space

P = {(q_{1}, \dots, q_{N}, \overset{q}{˙}_{1}, \dots, \overset{q}{˙}_{N}) : q_{i} \neq = q_{j} for i \neq = j} \subset R^{6 N}

. With respect to Lebesgue measure on

P

, the set of initial conditions whose maximal solution has a finite endpoint

T < \infty

while

i \neq = j min ∣ q_{i} (t) - q_{j} (t) ∣ \neq ⟶ 0 as t ↑ T

has measure zero.

DYN-015

Gottschalk Surjunctivity Conjecture

MajorConjecture1973

For every group

G

and every finite alphabet

A

, every continuous

G

-equivariant injective map

τ : A^{G} \to A^{G}

, with

A^{G}

carrying the product topology and the shift action, is surjective.

PROB-002

KPZ fluctuation exponent for two-dimensional first-passage percolation

LandmarkConjecturec. 1986

Give each unoriented nearest-neighbor edge

e

Z^{2}

an independent exponential random variable

τ_{e}

of mean

1

. For

x, y \in Z^{2}

, define

T (x, y) = π : x \to y min e \in π \sum τ_{e},

where the minimum is over finite nearest-neighbor paths from

x

y

. If

e_{1} = (1, 0)

, then

n \to \infty lim \frac{lo g Var T ( 0 , n e _{1} )}{lo g n} = \frac{2}{3} .

MATH-PHYS-008

Quantum unique ergodicity

LandmarkConjecture1994

Let

(M, g)

be a closed connected Riemannian manifold with strictly negative sectional curvature, and let

- Δ_{g} φ_{j} = λ_{j}^{2} φ_{j}

∥ φ_{j} ∥_{L^{2}} = 1

, with

λ_{j} \to \infty

. For every classical order-zero pseudodifferential operator

A \in Ψ^{0} (M)

⟨ A φ_{j}, φ_{j} ⟩ ⟶ \int_{S^{*} M} σ_{0} (A) d L .

Here

S^{*} M = {(x, ξ) \in T^{*} M : ∣ ξ ∣_{g} = 1}

σ_{0} (A)

is the degree-zero principal symbol restricted to

S^{*} M

, and

L

is normalized Liouville probability measure.